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


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WORST_CASE(?,O(n^1))
* Step 1: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(a(),b()) -> f(a(),c())
            f(c(),d()) -> f(b(),d())
        - Signature:
            {f/2} / {a/0,b/0,c/0,d/0}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,d,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(a) = [5]                  
            p(b) = [0]                  
            p(c) = [9]                  
            p(d) = [0]                  
            p(f) = [1] x1 + [1] x2 + [0]
          
          Following rules are strictly oriented:
          f(c(),d()) = [9]       
                     > [0]       
                     = f(b(),d())
          
          
          Following rules are (at-least) weakly oriented:
          f(a(),b()) =  [5]       
                     >= [14]      
                     =  f(a(),c())
          
        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:
            f(a(),b()) -> f(a(),c())
        - Weak TRS:
            f(c(),d()) -> f(b(),d())
        - Signature:
            {f/2} / {a/0,b/0,c/0,d/0}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,d,f}
    + Applied Processor:
        NaturalMI {miDimension = 2, 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) = [15]                     
                 [0]                      
          p(b) = [10]                     
                 [0]                      
          p(c) = [8]                      
                 [8]                      
          p(d) = [8]                      
                 [8]                      
          p(f) = [1 1] x1 + [1 0] x2 + [1]
                 [0 0]      [0 0]      [0]
        
        Following rules are strictly oriented:
        f(a(),b()) = [26]      
                     [0]       
                   > [24]      
                     [0]       
                   = f(a(),c())
        
        
        Following rules are (at-least) weakly oriented:
        f(c(),d()) =  [25]      
                      [0]       
                   >= [19]      
                      [0]       
                   =  f(b(),d())
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(a(),b()) -> f(a(),c())
            f(c(),d()) -> f(b(),d())
        - Signature:
            {f/2} / {a/0,b/0,c/0,d/0}
        - Obligation:
            innermost derivational complexity wrt. signature {a,b,c,d,f}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))