/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: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(b(x1)) -> b(a(x1))
            b(a(x1)) -> a(c(b(x1)))
        - Signature:
            {a/1,b/1} / {c/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c}
    + 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) = [1 1] x1 + [0]
                 [0 0]      [2]
          p(b) = [1 1] x1 + [0]
                 [0 0]      [2]
          p(c) = [1 0] x1 + [0]
                 [0 0]      [0]
        
        Following rules are strictly oriented:
        b(a(x1)) = [1 1] x1 + [2]
                   [0 0]      [2]
                 > [1 1] x1 + [0]
                   [0 0]      [2]
                 = a(c(b(x1)))   
        
        
        Following rules are (at-least) weakly oriented:
        a(b(x1)) =  [1 1] x1 + [2]
                    [0 0]      [2]
                 >= [1 1] x1 + [2]
                    [0 0]      [2]
                 =  b(a(x1))      
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(b(x1)) -> b(a(x1))
        - Weak TRS:
            b(a(x1)) -> a(c(b(x1)))
        - Signature:
            {a/1,b/1} / {c/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c}
    + 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(a) = [1 1] x1 + [0] 
                 [0 1]      [0] 
          p(b) = [1 2] x1 + [8] 
                 [0 1]      [12]
          p(c) = [1 0] x1 + [0] 
                 [0 0]      [0] 
        
        Following rules are strictly oriented:
        a(b(x1)) = [1 3] x1 + [20]
                   [0 1]      [12]
                 > [1 3] x1 + [8] 
                   [0 1]      [12]
                 = b(a(x1))       
        
        
        Following rules are (at-least) weakly oriented:
        b(a(x1)) =  [1 3] x1 + [8] 
                    [0 1]      [12]
                 >= [1 2] x1 + [8] 
                    [0 0]      [0] 
                 =  a(c(b(x1)))    
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(b(x1)) -> b(a(x1))
            b(a(x1)) -> a(c(b(x1)))
        - Signature:
            {a/1,b/1} / {c/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))