/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(b(x)) -> b(a(x))
            a(c(x)) -> x
        - Signature:
            {a/1} / {b/1,c/1}
        - Obligation:
             derivational complexity wrt. signature {a,b,c}
    + 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 + [0]
          p(b) = [1] x1 + [0]
          p(c) = [1] x1 + [7]
        
        Following rules are strictly oriented:
        a(c(x)) = [1] x + [7]
                > [1] x + [0]
                = x          
        
        
        Following rules are (at-least) weakly oriented:
        a(b(x)) =  [1] x + [0]
                >= [1] x + [0]
                =  b(a(x))    
        
* Step 2: NaturalMI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            a(b(x)) -> b(a(x))
        - Weak TRS:
            a(c(x)) -> x
        - 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 + [2]
                 [0 1]      [0]
          p(b) = [1 0] x1 + [0]
                 [0 1]      [1]
          p(c) = [1 6] x1 + [8]
                 [0 1]      [2]
        
        Following rules are strictly oriented:
        a(b(x)) = [1 1] x + [3]
                  [0 1]     [1]
                > [1 1] x + [2]
                  [0 1]     [1]
                = b(a(x))      
        
        
        Following rules are (at-least) weakly oriented:
        a(c(x)) =  [1 7] x + [12]
                   [0 1]     [2] 
                >= [1 0] x + [0] 
                   [0 1]     [0] 
                =  x             
        
* Step 3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            a(b(x)) -> b(a(x))
            a(c(x)) -> x
        - 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))