/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(b(b(a(x1))))
            b(b(b(x1))) -> b(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 + [0]
            p(b) = [1] x1 + [9]
          
          Following rules are strictly oriented:
          b(b(b(x1))) = [1] x1 + [27]
                      > [1] x1 + [18]
                      = b(b(x1))     
          
          
          Following rules are (at-least) weakly oriented:
          a(b(a(x1))) =  [1] x1 + [9]  
                      >= [1] x1 + [18] 
                      =  a(b(b(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(b(b(a(x1))))
        - Weak TRS:
            b(b(b(x1))) -> b(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 2]      [0]
                 [0 0 2] x1 + [0]
                 [0 0 0]      [2]
          p(b) = [1 0 2]      [0]
                 [0 0 1] x1 + [0]
                 [0 0 0]      [0]
        
        Following rules are strictly oriented:
        a(b(a(x1))) = [1 1 2]      [6]
                      [0 0 0] x1 + [0]
                      [0 0 0]      [2]
                    > [1 1 2]      [4]
                      [0 0 0] x1 + [0]
                      [0 0 0]      [2]
                    = a(b(b(a(x1))))  
        
        
        Following rules are (at-least) weakly oriented:
        b(b(b(x1))) =  [1 0 2]      [0]
                       [0 0 0] x1 + [0]
                       [0 0 0]      [0]
                    >= [1 0 2]      [0]
                       [0 0 0] x1 + [0]
                       [0 0 0]      [0]
                    =  b(b(x1))        
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(b(a(x1))) -> a(b(b(a(x1))))
            b(b(b(x1))) -> b(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))