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


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WORST_CASE(?,O(n^2))
* Step 1: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
            +(s(x),s(y)) -> s(+(s(x),+(y,0())))
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
             derivational complexity wrt. signature {+,0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = NoUArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          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 + [1]
            p(0) = [4]                  
            p(s) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
             +(0(),y) = [1] y + [5]
                      > [1] y + [0]
                      = y          
          
          +(s(x),0()) = [1] x + [5]
                      > [1] x + [0]
                      = s(x)       
          
          
          Following rules are (at-least) weakly oriented:
          +(s(x),s(y)) =  [1] x + [1] y + [1]
                       >= [1] x + [1] y + [6]
                       =  s(+(s(x),+(y,0())))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(s(x),s(y)) -> s(+(s(x),+(y,0())))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
        - 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 0] x1 + [1 2] x2 + [4]
                 [0 1]      [0 1]      [0]
          p(0) = [0]                      
                 [0]                      
          p(s) = [1 0] x1 + [0]           
                 [0 1]      [4]           
        
        Following rules are strictly oriented:
        +(s(x),s(y)) = [1 0] x + [1 2] y + [12]
                       [0 1]     [0 1]     [8] 
                     > [1 0] x + [1 2] y + [8] 
                       [0 1]     [0 1]     [8] 
                     = s(+(s(x),+(y,0())))     
        
        
        Following rules are (at-least) weakly oriented:
           +(0(),y) =  [1 2] y + [4]
                       [0 1]     [0]
                    >= [1 0] y + [0]
                       [0 1]     [0]
                    =  y            
        
        +(s(x),0()) =  [1 0] x + [4]
                       [0 1]     [4]
                    >= [1 0] x + [0]
                       [0 1]     [4]
                    =  s(x)         
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
            +(s(x),s(y)) -> s(+(s(x),+(y,0())))
        - 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))