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


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WORST_CASE(?,O(n^2))
* Step 1: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            f(+(x,0())) -> f(x)
        - Signature:
            {+/2,f/1} / {0/0}
        - Obligation:
            innermost derivational complexity wrt. signature {+,0,f}
    + 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 + [9]
            p(0) = [0]                  
            p(f) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          f(+(x,0())) = [1] x + [9]
                      > [1] x + [0]
                      = f(x)       
          
          
          Following rules are (at-least) weakly oriented:
          +(x,+(y,z)) =  [1] x + [1] y + [1] z + [18]
                      >= [1] x + [1] y + [1] z + [18]
                      =  +(+(x,y),z)                 
          
        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:
            +(x,+(y,z)) -> +(+(x,y),z)
        - Weak TRS:
            f(+(x,0())) -> f(x)
        - Signature:
            {+/2,f/1} / {0/0}
        - Obligation:
            innermost derivational complexity wrt. signature {+,0,f}
    + 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 1] x2 + [3]
                 [0 1]      [0 1]      [1]
          p(0) = [3]                      
                 [2]                      
          p(f) = [1 0] x1 + [0]           
                 [0 0]      [2]           
        
        Following rules are strictly oriented:
        +(x,+(y,z)) = [1 0] x + [1 1] y + [1 2] z + [7]
                      [0 1]     [0 1]     [0 1]     [2]
                    > [1 0] x + [1 1] y + [1 1] z + [6]
                      [0 1]     [0 1]     [0 1]     [2]
                    = +(+(x,y),z)                      
        
        
        Following rules are (at-least) weakly oriented:
        f(+(x,0())) =  [1 0] x + [8]
                       [0 0]     [2]
                    >= [1 0] x + [0]
                       [0 0]     [2]
                    =  f(x)         
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            f(+(x,0())) -> f(x)
        - Signature:
            {+/2,f/1} / {0/0}
        - Obligation:
            innermost derivational complexity wrt. signature {+,0,f}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))