/export/starexec/sandbox/solver/bin/starexec_run_tct_dc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/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(a(x)) -> b(b(x))
            b(b(a(x))) -> a(b(b(x)))
        - Signature:
            {a/1,b/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, 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] x1 + [8]
          p(b) = [1] x1 + [1]
        
        Following rules are strictly oriented:
        a(a(x)) = [1] x + [16]
                > [1] x + [2] 
                = b(b(x))     
        
        
        Following rules are (at-least) weakly oriented:
        b(b(a(x))) =  [1] x + [10]
                   >= [1] x + [10]
                   =  a(b(b(x)))  
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            b(b(a(x))) -> a(b(b(x)))
        - Weak TRS:
            a(a(x)) -> b(b(x))
        - Signature:
            {a/1,b/1} / {}
        - Obligation:
             derivational complexity wrt. signature {a,b}
    + 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 6] x1 + [0]
                 [0 1]      [2]
          p(b) = [1 2] x1 + [0]
                 [0 1]      [0]
        
        Following rules are strictly oriented:
        b(b(a(x))) = [1 10] x + [8]
                     [0  1]     [2]
                   > [1 10] x + [0]
                     [0  1]     [2]
                   = a(b(b(x)))    
        
        
        Following rules are (at-least) weakly oriented:
        a(a(x)) =  [1 12] x + [12]
                   [0  1]     [4] 
                >= [1 4] x + [0]  
                   [0 1]     [0]  
                =  b(b(x))        
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(a(x)) -> b(b(x))
            b(b(a(x))) -> a(b(b(x)))
        - Signature:
            {a/1,b/1} / {}
        - Obligation:
             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^2))