/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^1))
* Step 1: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(b(a(x1))) -> a(a(b(b(a(a(x1))))))
            b(a(a(b(x1)))) -> b(a(b(x1)))
        - Signature:
            {a/1,b/1} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b}
    + 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(a) = [1] x1 + [2]
            p(b) = [1] x1 + [1]
          
          Following rules are strictly oriented:
          b(a(a(b(x1)))) = [1] x1 + [6]
                         > [1] x1 + [4]
                         = b(a(b(x1))) 
          
          
          Following rules are (at-least) weakly oriented:
          a(b(a(x1))) =  [1] x1 + [5]        
                      >= [1] x1 + [10]       
                      =  a(a(b(b(a(a(x1))))))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 2: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            a(b(a(x1))) -> a(a(b(b(a(a(x1))))))
        - Weak TRS:
            b(a(a(b(x1)))) -> b(a(b(x1)))
        - Signature:
            {a/1,b/1} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b}
    + Applied Processor:
        NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = NoUArgs, urules = NoURules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind triangular matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
          p(a) = [1 1 0]      [0]
                 [0 0 0] x1 + [0]
                 [0 0 0]      [1]
          p(b) = [1 0 1]      [0]
                 [0 0 1] x1 + [0]
                 [0 0 0]      [0]
        
        Following rules are strictly oriented:
        a(b(a(x1))) = [1 1 0]      [2]    
                      [0 0 0] x1 + [0]    
                      [0 0 0]      [1]    
                    > [1 1 0]      [1]    
                      [0 0 0] x1 + [0]    
                      [0 0 0]      [1]    
                    = a(a(b(b(a(a(x1))))))
        
        
        Following rules are (at-least) weakly oriented:
        b(a(a(b(x1)))) =  [1 0 2]      [1]
                          [0 0 0] x1 + [1]
                          [0 0 0]      [0]
                       >= [1 0 2]      [1]
                          [0 0 0] x1 + [1]
                          [0 0 0]      [0]
                       =  b(a(b(x1)))     
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(b(a(x1))) -> a(a(b(b(a(a(x1))))))
            b(a(a(b(x1)))) -> b(a(b(x1)))
        - Signature:
            {a/1,b/1} / {}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))